Video Transcript
Complete the following sentence. An angular displacement of 0.45 radians is equal to blank revolutions.
In this question, what we essentially want to do is to convert an angle in radians into a number of revolutions. We can be helped in doing this by recalling that if we have a circular arc and we move once around that arc, that means we’ve passed through one complete revolution, or equivalently two 𝜋 radians. We can write then that one revolution equals two 𝜋 radians, and therefore one revolution divided by two 𝜋 radians is equal to one. Since this fraction is equal to one, we can multiply a quantity by it without essentially changing that quantity.
If we take our given angle of 0.45 radians and multiply it by one revolution per two 𝜋 radians, since we are multiplying by one, we don’t essentially change this quantity. But notice what change does take place. The units of radians in numerator and denominator cancel out. And what we’ve done is effectively converted this angle in radians into a measure of revolutions. 0.45 times one divided by two 𝜋 to three decimal places is 0.072 with units of revolutions. Our completed sentence then reads “An angular displacement of 0.45 radians is equal to 0.072 revolutions.”
